A cycle that contains finitely many elements is a *finite cycle* and the
*cycle length* is the number of elements it contains.

Let be a permutation written as a finite product of disjoint cycles of finite length. The order of is the least common multiple of the lengths of the cycles.

A *transposition* is a cycle of length two. Clearly a transposition
has order two, but there are permutations of order two that are not transpositions.

One important application of transpositions is that
every permutation may be written as a product of transpositions (although not
necessarily disjoint and not uniquely).
A permutation is an *even permutation* if it is a product of an even number of
transpositions and an *odd permutation* if it is a product of an odd number of
transpositions.

As the definition suggests, a permutation cannot be both odd and even. However, the decomposition into a product of transpositions is not unique nor is the number of transpositions unique. For example, the permutation may be written as , or . Of course, the identity permutation is an even permutation.

Sun Mar 30 14:48:35 PST 1997